On the space of subgroups of Baumslag-Solitar groups II: High transitivity

TL;DR

This paper explores the conditions under which Baumslag-Solitar groups exhibit highly transitive actions, revealing that high transitivity is generic in certain phenotypes and characterizing the dynamical properties of these actions.

Contribution

It extends previous work by analyzing high transitivity in the space of actions, identifying when such actions are generic or absent based on the phenotype of the group actions.

Findings

High transitivity is generic for phenotype 1 actions.

High transitivity is also generic in infinite phenotype cases.

High transitivity does not occur when |m|=|n| in infinite phenotype.

Abstract

We continue our study of the perfect kernel of the space of transitive actions of Baumslag-Solitar groups by investigating high transitivity. We show that actions of finite phenotype are never highly transitive, except when the phenotype is 1, in which case high transitivity is actually generic. In infinite phenotype, high transitivity is generic, except when |m|=|n| where it is empty. We also reinforce the dynamical properties of the action by conjugation on the perfect kernel that we had established in our first paper, replacing topological transitivity by high topological transitivity.